Defining custom Pareto evaluation functions

Defining Custom Pareto-evaluation Functions

PARODIS uses function handles to the four main types of Pareto functions. These are:

• Extreme Point Functions: Functions for finding the points that minimize exactly one objective
• Optimizer Functions: Functions for defining the optimizer used in the Front Determination Scheme
• Front Determination Schemes: Functions for calculating the Pareto front
• Metric Functions: Functions for selecting a Pareto-optimal point as a solution

Defining Extreme Point Functions

[extremePoints, inputsEP, slacksEP, parametersEP] = initializeExtremePointFunctionName(paretoObj, optimizeConstraints, costExpressions, agent)

Extreme Point Functions are meant to calculate the initial points of the Pareto-evaluation, output them with their respective inputs, slacks and paretoParameters, and set Utopia and Nadir points by any means.

Input	Type	Description
paretoObj	ParetoController	ParetoController object for setting Utopia and Nadir point.
optimizeConstraints	YALMIP Constraint	Constraints the system is subject to.
costExpressions	cell array	Cell array with cost functions.
agent	Agent	Calling Agent object.

Output	Type	Description
inputs	n-by-1 cell array	Saves the output u of every extreme point evaluation as a cell
paretoParameters	matrix	Matrix with the paretoParameters for finding each extreme point as a row vector

Defining Optimizer Functions

optimizer = prepareOptimizerMethodName(paretoObj, optimizeConstraints, costExpressions, agent, extremePoints)

Optimizer Functions output the optimizer that is used in the Front Determination method. Optimizer objects are YALMIP objects that presolve the optimization problem. They take numerical values and replace preset symbols with these. For more information regard https://yalmip.github.io/command/optimizer/. These functions take the system's constraints and scalar cost expressions to formulate the scalarization.

Input	Type	Description
paretoObj	ParetoController	ParetoController object for accessing config, symbolic weights, etc.
optimizeConstraints	sdpvar	Constraint output by ParetoController.prepareProblem().
costExpressions	sdpvar cell array	Cost expressions resulting from ParetoController.prepareProblem().
agent	Agent	Agent object for accessing model and agent config.
extremePoints	k-by-k matrix	Optional: Extreme points resulting from the extreme point function.

Output	Type	Description
optimizer	optimizer	Optimizer object used in the front determination scheme

Defining Front Determination Schemes

[inputs, slacks, front, parameters] = determinefrontDeterminationSchemeName( paretoObj, agent, optimizer, extremePoints, preselectedParameters )

Front Determination Schemes are functions that calculate the Pareto front by any means and output the newfound points and their respective inputs, slacks and paretoParameters. It can also be useful to update the Nadir point for better normalization.

Input	Type	Description
paretoObj	ParetoController	ParetoController object for accessing config, status, etc.
agent	Agent	Agent object for accessing model and agent config.
optimizer	optimizer	Optimizer for calculating the new solution by inputing paretoParameters.
extremePoints	k-by-k matrix	Extreme points resulting from the extreme point method.
preselectedParameters	k-by-1 vector	Optional: A preset paretoParameters vector to calculate the corresponding solution for.

Output	Type	Description
inputs	m-by-1 cell array	Saves the output u of every Pareto-evaluation as a cell array.
slacks	m-by-1 cell array	Saves the slack variable values of every Pareto-evaluation as a cell array.
front	m-by-k matrix	Matrix with Pareto optimal points as rows.
paretoParameters	matrix	Matrix with the paretoParameters of each Pareto-optimal point as a row vector.

Defining Metric Functions

[idx, utility] = selectMetricFunctionName( varargin )

Metric functions are used for automatically selecting a Pareto-optimal point on the Pareto front. They can be used by either inputting a Pareto front to calculate the metric for or by inputting an agent object with a timestep and selecting the corresponding Pareto front.

Input	Type	Description
agentObj	Agent	Agent object where the Pareto front is taken from the history.
timeStep	integer	Timestep to load the Pareto front from.

OR

Input	Type	Description
Pareto front	m-by-k matrix	Pareto front with points as row vectors

Output	Type	Description
idx	integer	Index of the chosen Pareto-optimal point

Naming Convention

The predefined methods are accessible by setting the corresponding config parameter to a string with matching name. Users can expand this set of strings by following the naming convention for these methods and adding the new function to the ParetoController.

Method type	Prefix	Example
Extreme Point Function	initialize	initializeOAA()
Optimizer Methods	prepare	prepareNBI()
Front Determination Schemes	determine	determineNBI()
Metric Function	select	selectCUP()